A Non-Cooperative Support for Equal Division
in Estate Division Problems∗
Itai Ashlagi†
Emin Karagözoğlu‡
Bettina Klaus§
Final Version Mathematical Social Sciences: January 2012
Abstract
We consider estate division problems and show that for any claim game based on a (estate
division) rule satisfying efficiency, equal treatment of equals, and order preservation of awards,
all (pure strategy) Nash equilibria induce equal division. Next, we consider (estate division)
rules satisfying efficiency, equal treatment of equals, and claims monotonicity. Then, for claim
games with at most three agents, again all Nash equilibria induce equal division. Surprisingly,
this result does not extend to claim games with more than three agents. However, if nonbossiness
is added, then equal division is restored.
JEL classification: C72, D63, D71.
Keywords: Bankruptcy/estate division problems, claims monotonicity, direct revelation claim
game, equal division, equal treatment of equals, Nash equilibria, nonbossiness, order preservation
(of awards).
1
Introduction
We consider estate division problems, a generalization of bankruptcy problems, in which a positivevalued estate has to be divided among a set of agents. Clearly, if the agents’ claims add up to less
than the estate, no conflict occurs and each agent can receive his claimed amount. However, if the
sum of the agents’ claims exceeds the estate, then bankruptcy occurs. The class of bankruptcy
problems has been extensively studied using various approaches such as the normative (axiomatic)
or the game-theoretical approach (cooperative or noncooperative). For extensive surveys of the
literature, we refer to Moulin (2002) and Thomson (2003).
In bankruptcy problems the agents’ claims are normally considered as fixed inputs to the problem. However, in many real life situations it is impossible or difficult to check the validity of claims,
e.g., if the profit of a joint project should be split among the project participants, but inputs are not
perfectly observable or difficult to compare. Other examples are claims based on moral property
∗
The authors thank Serhat Doğan, Tarık Kara, Flip Klijn, Juan D. Moreno-Ternero, and William Thomson for
helpful comments. B. Klaus thanks the Netherlands Organisation for Scientific Research (NWO) for its support under
grant VIDI-452-06-013.
†
MIT Sloan School of Management, 50 Memorial Drive, Cambridge, Massachusetts 02142, USA; e-mail:
[email protected]
‡
Department of Economics, Bilkent University, 06800 Ankara, Turkey; e-mail: [email protected]
§
Corresponding Author: Faculty of Business and Economics, University of Lausanne, Internef 538 CH-1015 Lausanne, Switzerland; e-mail: [email protected].
1
rights, entitlements (see Gächter and Riedl, 2005) or subjective needs (see Pulido, Sanchez-Soriano,
and Llorca, 2002). If the authority in charge of the estate lacks the ability to verify claims or verification is too costly, agents are likely to behave strategically to ensure larger shares of the estate
for themselves.
We model this type of situation with a simple noncooperative game that resembles Nash’s
classical demand game (Nash, 1953). Given the estate to divide and based on a (estate division)
rule, agents simply submit claims which are restricted to not exceed a common upper bound. We
analyze the (pure strategy) Nash equilibria of the resulting claim game. We do not fix any specific
rule, but only require the rule to satisfy basic and appealing properties.
First, we require the rule to satisfy efficiency, equal treatment of equals, and order preservation
of awards.1 Then, all agents claiming the largest possible amount is a Nash equilibrium and all Nash
equilibria lead to equal division (Theorem 1). Second, we replace order preservation of awards with
claims monotonicity.2 Again, all agents claiming the largest possible amount is a Nash equilibrium.
However, in contrast to the previous result, we show that equal division is guaranteed for all Nash
equilibria only for claim games with at most three agents (Theorem 2). Surprisingly, this result does
not extend to claim games with more than three agents (Example 1). Nevertheless, if nonbossiness
is added, then equal division in all Nash equilibria is restored (Theorem 3).3
All our results point towards the same intuitive message: if it is impossible or difficult to test
the legitimacy of claims, the conflict will escalate to the highest possible level at which claims
are no longer informative. As a result, equal division is the “non-discriminating” outcome in Nash
equilibrium. In other words, equal division is not only a normatively appealing division method, but
it is also the result of a natural noncooperative game.4 These findings might explain why in certain
instances equal division is applied right away even without asking agents’ claims. For instance, pre1975 U.S. Admiralty law divides liabilities equally among parties if they are both found negligent
(see Feldman and Kim, 2005). British Shipping Law, until the act of 1911, applied equal division of
costs in case of a collision between two ships, however much the degree of their faults or negligence
may differ. This practice has originated from a medieval rule, which was originally intended to be
applied only in cases where negligence cannot be perfectly proven (see Porges and Thomas, 1963).
Finally, in Arizona, California, Idaho, Louisiana, Nevada, New Mexico, Texas, Washington, and
Wisconsin, “community property law” implements equal split of assets and wealth accumulated
during marriage in case of a divorce.
A number of articles also consider strategic aspects of claim problems (see Thomson, 2003,
Section 7). The articles closely related to ours are Chun (1989), Moreno-Ternero (2002), Herrero
(2003), and Bochet, Sakai, and Thomson (2010) in that the games they consider do not focus on a
specific rule, but a class of rules that is determined by basic properties. Chun (1989) considers a
noncooperative game where agents propose rules and a sequential revision procedure then converges
to equal division. Moreno-Ternero (2002) constructs a noncooperative game, the equilibrium of
which converges to the proportional rule. A noncooperative game similar (in a sense dual) to the
1
Efficiency: the estate is allocated if the sum of claims is larger than (or equal to) the estate. Equal treatment of
equals: any two agents with identical claims receive the same awards. Order preservation of awards: if an agent has
a higher claim than another agent, then he does not receive less than that agent.
2
Claims monotonicity: other things equal, an agent does not receive less after an increase in his claim.
3
Nonbossiness: no agent can change other agents’ awards by changing his claim unless his award changes as well.
4
A game-theoretical interpretation of our result is that if a rule satisfies certain natural and appealing properties
(see our results above), it can be used to implement equal division.
2
one in Chun (1989) is constructed by Herrero (2003) who shows convergence to the constrained
equal losses rule. Bochet, Sakai, and Thomson (2010) consider the problem of allocating an estate
when agents have single-peaked preferences and study direct revelation games associated with
(allocation) rules. They prove that uniform division is the only Nash equilibrium outcome for
rules satisfying certain properties (Bochet, Sakai, and Thomson, 2010, Theorem 2).5 Note that
uniform division (for single-peaked preferences) is similar in spirit to equal division (for monotone
preferences) in that a uniform allocation divides the estate as equally as possible by either taking
agents’s peaks as upper bounds (in case of bankruptcy) or as lower bounds (in case of an excess
supply). Hence, in this paper as well as in Bochet, Sakai, and Thomson (2010), based on agents
preferences, we implement the appropriate notion of equality in allocation.6
The paper is organized as follows. In Section 2, we introduce claim games as well as various
properties for underlying rules and three well-known rules (the proportional, the constrained equal
awards, and the constrained equal losses rule). In Section 3, we establish our equal division Nash
equilibria results (Theorems 1, 2, and 3, and Example 1) and discuss the independence of assumptions needed to establish our results (Remarks 1, 2, and 3), the relation between claim games
and divide-the-dollar games (Remark 4), and the Nash equilibria obtained for the proportional,
the constrained equal awards, and the constrained equal losses rule (Remark 5). We conclude in
Section 4.
2
The claim game
In a claim game, an estate E > 0 has to be divided among a set of agents N = {1, . . . , n}. We
assume that agents’ preferences are strictly monotone over the amounts of the estate they receive.
Then, the estate E and the set of agents N determine an estate division problem.
A strategy for an agent i ∈ N is a claim ci ≥ 0 belonging to her non-empty strategy set
Ci = [0, k] (k > 0). For example, we could assume for all i ∈ N , Ci = [0, E]. The set of strategy
profiles (claims vectors) is denoted by C = C1 × . . . × Cn . Hence, for all i ∈ N , the maximal claim
c̄i ≡ max Ci = k and the maximal claims vector c̄ ≡ (c̄i )i∈N . We assume that the estate E, the set
of agents N , and the set of strategy profiles C are fixed.
We
each c ∈ C and each S ⊆ N , S 6= ∅, let
Puse the following notations in the sequel. For
0
cS = i∈S ci . For each c ∈ C, each i ∈ N , and each ci ∈ Ci , (c0i , c−i ) ∈ C denotes the claims vector
obtained from c by replacing ci with c0i .
N
A claim game’s outcome function is a (estate division) rule R : C
P→ R+ that associates with7
N
each strategy profile c ∈ C an awards vector R(c) ∈ R+ such that
Ri (c) ≤ E and R(c) ≤ c;
Ri (c) denotes the amount of estate E that agent i obtains under strategy profile c. We do not fix
the rule that determines the outcomes of a claim game and therefore denote a claim game by Γ(R).
Note that the rules that determine the outcomes of our claim games can be thought of as
extended bankruptcy rules (see Thomson, 2003, for a comprehensive survey on the axiomatic and
game-theoretic analysis of bankruptcy problems). We now introduce some properties of rules. All
properties are stated for a generic rule R.
5
The properties they consider – peak-only, efficiency, symmetry, others-oriented peak monotonicity, peak continuity, and strict own-peak monotonicity – are similar in spirit to the ones we consider for estate division problems.
6
Intuitively speaking, Bochet, Sakai, and Thomson (2010) require more properties to obtain their result to accommodate the role agents’ peaks play as upper or lower bounds when allocating the estate as equally as possible.
7
Note that R(c) ≤ c if and only if for all i ∈ N , Ri (c) ≤ ci .
3
Efficiency: the largest possible
amount of E is assigned taking claims as upper bounds, i.e., for
P
all c ∈ C, [if cN ≥ E, then
R(c)i = E] and [if cN ≤ E, then R(c) = c].
Note that we do not require that E has to be completely allocated among the agents if no
bankruptcy occurs (E < cN ).8
The following property requires that the awards to agents whose claims are equal should be
equal.
Equal treatment of equals: for all c ∈ C and all i, j ∈ N such that ci = cj , Ri (c) = Rj (c).
By the next requirement, the ordering of awards should conform to the ordering of claims, i.e.,
if agent i’s claim is larger than agent j’s claim, i should receive at least as much as agent j does.
Order preservation of awards: for all c ∈ C and all i, j ∈ N such that ci > cj , Ri (c) ≥ Rj (c).
Order preservation of awards is a weakening of the standard order preservation property introduced by Aumann and Maschler (1985) (which additionally also requires order preservation of
losses).9
The following monotonicity property requires that, other things equal, if an agent’s claim increases, he should receive at least as much as he did initially.
Claims monotonicity: for all c ∈ C, all i ∈ N , and all c0i ∈ Ci such that ci < c0i , Ri (c) ≤
Ri (c0i , c−i ).
Most well-known bankruptcy rules satisfy all the properties mentioned above; e.g., the constrained equal awards, the constrained equal losses, and the proportional rule. We introduce efficient extensions of these well-known bankruptcy rules to the (estate division) rules that serve as
outcome functions for our claim games.
The constrained equal awards rule allocates the estate as equally as possible taking claims as
upper bounds.
Constrained equal awards rule, CEA. For each c ∈ C,
(i) if cN ≤ E, then CEA(c) = c and
(ii) if cN ≥ E, then for all jP
∈ N , CEAj (c) = min{cj , λcea },
where λcea is such that
min{ci , λcea } = E.
The constrained equal losses rule allocates the shortage of the estate in an equal way, keeping
awards bounded below by zero.
Constrained equal losses rule, CEL. For each c ∈ C,
(i) if cN ≤ E, then CEL(c) = c and
(ii) if cN ≥ E, then for all jP
∈ N , CELj (c) = max{0, cj − λcel },
where λcel is such that
max{0, ci − λcel } = E.
The proportional rule allocates the estate proportionally with respect to claims.
Proportional rule, P. For each c ∈ C,
(i) if cN ≤ E, then P (c) = c and
(ii) if cN ≥ E, then P (c) = λp c, where λp =
E
cN .
8
In Appendix B we describe what happens if we require that the estate E is always completely allocated among
the agents. Then, efficiency is already incorporated in the definition of a rule and results essentially do not change.
9
Order preservation. A rule R satisfies order preservation if for all c ∈ C and all i, j ∈ N such that ci ≥ cj ,
Ri (c) ≥ Rj (c) (order preservation of awards) and ci − Ri (c) ≥ cj − Rj (c) (order preservation of losses).
4
3
Nash equilibria and equal division
First, we are interested in (pure strategy) Nash equilibria of the claim game. A claims vector
c ∈ C is a Nash equilibrium (in pure strategies) of claim game Γ(R) if for all i ∈ N and all c0i ∈ Ci ,
Ri (c) ≥ Ri (c0i , c−i ); we call R(c) the Nash equilibrium outcome. Since agents’ preferences are strictly
monotone over the amounts of the estate they receive, any Nash equilibrium of a claim game that
is based on an efficient rule has to distribute the whole estate if that is possible given upper bounds
c̄ on reported claims vectors. This implies that at any Nash equilibrium c at which agents do not
claim their maximal possible amounts (c 6= c̄), the sum of reported claims must add up to at least
the estate (cN ≥ E).
Lemma 1. If rule R is efficient, then for any Nash equilibrium c of the claim game Γ(R), c 6= c̄
implies cN ≥ E.
Proof. Let rule R be efficient and assume that c 6= c̄ is a Nash equilibrium of the claim game
Γ(R) such that cN < E. Let α ≡ E − cN > 0 and define for some j ∈ N such that cj < c̄j ,
c0j ≡ min{c̄j , cj + α} > cj . Then, by efficiency, R(c) = c and R(c0j , c−j ) = (c0j , c−j ). Hence,
Rj (c) = cj < c0j = Rj (c0i , c−i ); contradicting that c is a Nash equilibrium of Γ(R).
We denote by 1 = (1, . . . , 1) ∈ RN
++ the one-vector.
Equal division. Given an estate E > 0,
vector.
E
n1
∈ RN
++ denotes the corresponding equal division
Next, we show that for claim games where agents have equal maximal strategies and the underlying rule satisfies efficiency, equal treatment of equals, and order preservation of awards, (a)
claiming the maximal amount is always a Nash equilibrium and (b) all Nash equilibria induce equal
division.
Theorem 1. Let rule R satisfy efficiency, equal treatment of equals, and order preservation of
awards. Then,
(a) the maximal claims vector c̄ is a Nash equilibrium of the claim game Γ(R) and
(b) the outcome in all Nash equilibria of the claim game Γ(R) is min{k, En }1.
Proof.
(a) We prove that c̄ = k1 is a Nash equilibrium of the claim game Γ(R). By efficiency and equal
treatment of equals, R(c̄) = min{k, En }1. If R(c̄) = k1, then each agent already gets the largest
possible amount and c̄ is a Nash equilibrium.
Thus, assume that R(c̄) = En 1 <k1. Let i ∈ N and c0i 6= c̄i . Thus, for all j 6= i, c0i < k = c̄j .
Hence, by order preservation of awards, for all j 6= i,PRj (c0i , c̄−i ) ≥ Ri (c0i , c̄−i ). Suppose that
Ri (c0i , c̄−i ) > En . Then, for all l ∈ N , Rl (c0i , c̄−i ) > En and
Rl (c0i , c̄−i ) > E; a contradiction. Thus,
E
0
Ri (ci , c̄−i ) ≤ n = Ri (c̄) and c̄ is a Nash equilibrium of the claim game Γ(R).
(b) Suppose that c is a Nash equilibrium of the claim game Γ(R) and R(c) 6= min{k, En }1. Then,
for some i ∈ N , Ri (c) < min{k, En }. Let c0i = k (possibly c0i = ci ). Since c is a Nash equilibrium,
Ri (c0i , c−i ) ≤ Ri (c) < min{k, En }. In particular, (i) Ri (c0i , c−i ) < En .
E
Since by (a), c̄ is a Nash equilibrium such that R(c̄) = min{k,
n }1, we know that c 6= c̄. Thus,
P
0
0
by
1, cN ≥ E. Recall that ci = k ≥ ci . Hence, ci + l6=i cl ≥ E and by efficiency,(ii)
P Lemma
Rl (c0i , c−i ) = E. For all j 6= i such that cj < k = c0i , by order preservation of awards and (i),
i such that cj = k = c0i , by equal treatment of equals and
Rj (c0i , c−i ) ≤ Ri (c0i , c−i ) < En . For all j 6= P
E
0
0
(i), Rj (ci , c−i ) = Ri (ci , c−i ) < n . Hence,
Rl (c0i , c−i ) < E; a contradiction to (ii).
5
Remark 1. Independence of assumptions in Theorem 1
(i) Suppose that not all agents have equal maximal claims, i.e., there exist i, j ∈ N such that
c̄i 6= c̄j . Then, an “unequal” Nash equilibrium is possible even if the rule satisfies efficiency, equal
treatment of equals, and order preservation of awards; e.g., for the proportional rule c̄ is a Nash
equilibrium, but for all i, j such that c̄i 6= c̄j , Pi (c̄) 6= Pj (c̄).
(ii) The following rule R satisfies equal treatment of equals, order preservation of awards, but not
efficiency. If c 6= c̄, then R(c) = P (c) and R(c̄) = 01. Clearly, c̄ is not a Nash equilibrium of the
claim game Γ(R) and the equal division vector min{k, En }1 is never an equilibrium outcome.
(iii) A serial dictatorship rule that first serves agents with the highest claims lexicographically (i.e.,
if several agents have the highest claim, then first serve the agent with the lowest index and so on)
satisfies efficiency and order preservation of awards, but not equal treatment of equals. There are
Nash equilibria, e.g., c = k1 when nk > E, at which agent 1 receives more than agent n.
(iv) The following rule R satisfies efficiency and equal treatment of equals, but not order preservation of awards. Rule R first assigns the estate E proportionally (and efficiently) among all agents
who have a claim different from that of agent 1. Then, if some part of the estate is left, R allocates
it equally (and efficiently) among the remaining agents. For c̄ 6= E1, c̄ is not a Nash equilibrium
of the claim game Γ(R) and the equal division vector min{k, En }1 is not an equilibrium outcome. In Ashlagi, Karagözoğlu, and Klaus (2008, Corollaries 1 and 2) two corresponding results are
obtained using order preservation9 (instead of equal treatment of equals and order preservation of
awards) or others oriented claims monotonicity 10 (instead of order preservation of awards).
Next, we show that for claim games where agents have equal maximal strategies and the underlying rule satisfies efficiency, equal treatment of equals, and claims monotonicity, (a) claiming the
maximal amount is always a Nash equilibrium and (b) for n ≤ 3, all Nash equilibria induce equal
division.
Theorem 2. Let rule R satisfy efficiency, equal treatment of equals, and claims monotonicity.
Then,
(a) the maximal claims vector c̄ is a Nash equilibrium of the claim game Γ(R) and
(b) for n ≤ 3, the outcome in all Nash equilibria of the claim game Γ(R) is min{k, En }1.
Proof.
(a) By claims monotonicity, for each agent i it is a weakly dominant strategy to claim c̄i . Hence,
c̄ is a Nash equilibrium of Γ(R).
(b) For n = 1, the proof is obvious and therefore omitted.
For n = 2, Ashlagi, Karagözoğlu, and Klaus (2008, Lemma 3) prove that efficiency, equal
treatment of equals, and claims monotonicity imply order preservation of awards. By Theorem 1
(b), efficiency, equal treatment of equals, and order preservation of awards imply the result.
Let n = 3. Suppose that c is a Nash equilibrium of the claim game Γ(R) and R(c) 6= min{k, E3 }1.
Without loss of generality, we assume that c1 ≤ c2 ≤ c3 .
Case 1. i ∈ {1, 2} ≡ {i, j} and Ri (c) < min{k, E3 }.
Let c0i = c3 (possibly c0i = c1 ). Since c is a Nash equilibrium, Ri (c0i , c−i ) ≤ Ri (c) < min{k, E3 }. In
particular, (i) Ri (c0i , c−i ) < E3 .
Others oriented claims monotonicity: for all c ∈ C and all i ∈ N such that ci < c0i , Rj (c) ≥ Rj (c0i , c−i ) for
all j 6= i.
10
6
Since by (a), c̄ is a Nash equilibrium
such that R(c̄) = min{k, En }1,Pwe know that c 6= c̄. Thus,
P
by Lemma 1, cN ≥ E. Hence, c0i + l6=i cl ≥ E and by efficiency, (ii)
Rl (c0i , c−i ) = E. By equal
E
treatment of equals and (i), R3 (c0i , c−i ) = Ri (c0i , c−i ) < 3 . Hence, (ii) implies (iii) Rj (c0i , c−i ) > E3 .
Recall that j ∈ {1, 2} and therefore, cj ≤ c3 . Let c0j = c3 and consider (c0i , c0j , c3 ) = (c3 1). By
equal treatment of equals, (i) and (iii) imply cj < c3 . Hence, c0i + c0j + c3 > E and by efficiency,
P
(iv)
Rl (c0i , c0j , c3 ) = E. By claims monotonicity, Rj (c0i , c0j , c3 ) ≥ Rj (c0i , c−i ) > E3 and by equal
P
treatment of equals, Rj (c0i , c0j , c3 ) = Ri (c0i , c0j , c3 ) = R3 (c0i , c0j , c3 ) > E3 . Hence,
Rl (c0i , c0j , c3 ) > E;
a contradiction to (iv).
Case 2. R3 (c) < min{k, E3 }.
First, R3 (c) < min{k, E3 } implies (v) R3 (c) < E3 . Furthermore, if c2 = c3 , then by equal treatment
of equals, R2 (c) = R3 (ĉ) < min{k, E3 }, and we are done by Case 1. Hence, assume that c2 < c3 .
Let c03 = c2 and consider (c03 , c−3 ).
E
Since by (a), c̄ is a Nash equilibrium such that R(c̄)
P = min{k, n }1, we know that c 6= c̄.
Thus, by Lemma 1, cN ≥ E. Hence, by efficiency, (vi)
Rl (c) = E. Then, (v) and (vi) imply
E
E
R1 (c) > 3 or R2 (c)P> 3 . Note that R1 (c) ≤ c1 ≤ c2 and R2 (c) ≤ c2 . Thus, c03 > R3 (c)
0
and
Rl (c) = E, where the last equality follows from (vi). By efficiency, (vii)
P c1 0+ c2 + c3 ≥
Rl (c3 , c−3 ) = E. By claims monotonicity, (viii) R3 (c03 , c−3 ) ≤ R3 (c) < E3 . By equal treatment
of equals, R2 (c03 , c−3 ) = R3 (c03 , c−3 ) < E3 . Hence, (vii) implies (ix) R1 (c03 , c−3 ) > E3 .
Let c01 = c2 and consider (c01 , c03 , c2 ) = (c2 1). By equal treatment
(viii) and (ix) imply
P of equals,
0
0
0
0
0
c1 < c3 = c2 . Hence, c1 + c3 + c2 > E and by efficiency, (x)
Rl (c1 , c3 , c2 ) = E. By claims
monotonicity, R1 (c01 , c03 , c2 ) ≥ R1 (c03 , c−3 ) P
> E3 and by equal treatment of equals, R1 (c01 , c03 , c2 ) =
R2 (c01 , c03 , c2 ) = R3 (c01 , c03 , c2 ) > E3 . Hence,
Rl (c01 , c03 , c2 ) > E; a contradiction to (x).
Remark 2. Independence of assumptions in Theorem 2
Note that claims monotonicity alone implies Theorem 2 (a). Hence, we show independence only
for Theorem 2 (b).
(i) The proof that it is essential that all agents have equal maximal claims is the same as in
Remark 1 (i).
(ii) The following rule R satisfies claims monotonicity and equal treatment of equals, but not
efficiency. If at c exactly one agent i claims ci = c̄i , then he receives Ri (c) = min{k, En } and
for all j 6= i, Ri (c) = 0. Furthermore, R(c̄) = min{k, En }1 and for all other claims vectors c,
R(c) = 01. Then, Nash equilibria which do not induce equal division exist, e.g., for N = {1, 2, 3}
and E = k = 1, c̃ = (1, 0, 0) resulting in the equilibrium outcome R(c) = (1/3, 0, 0).
(iii) To prove that equal treatment of equals is needed one can use the serial dictatorship rule as
described in Remark 1 (iii) – it satisfies efficiency and claims monotonicity, but not equal treatment
of equals.
(iv) To prove that claims monotonicity is needed one can use the rule as described in Remark 1 (iv)
– it satisfies efficiency and equal treatment of equals, but not claims monotonicity.
In the following example, we show that when n > 3, efficiency, equal treatment of equals, and
claims monotonicity are not sufficient to guarantee equal division in all Nash equilibria of the claim
game Γ(R). This example represents a claim game with an asymmetric equilibrium despite the
fact that players enter the game in a symmetric way. Note that all players have the same maximal
claims, the rule used satisfies equal treatment of equals, and yet there is an equilibrium at which
players receive unequal payoffs.
7
Example 1. Let N = {1, 2, 3, 4}, E = 1, and for all i ∈ N , Ci = [0, 1]. Before we define rule R of
claim game Γ(R), we introduce some notation.
Let H = 1/2 and L = 1/3 be two points, which we will use to partition the set of claim profiles.
For all profiles c ∈ C, let LH(c) = {i ∈ N : L ≤ ci ≤ H}, L(c) = {i ∈ N : ci < L}, and
H(c) = {i ∈ N : ci > H}. For all j = 0, 1, . . . , 4, denote by C j the set of claim profiles in which
j agents claim between L and H and n − j agents claim more than H. That is, C j = {c ∈ C :
|LH(c)| = j and |H(c)| = n − j}. Let P = C \ ∪4j=0 C j . Note that for all claim profiles c ∈ P ,
there exists some agent i ∈ N for which ci < L. Furthermore, note that the collection of sets C j
(j = 1, . . . , 4) and P partition the set of claim profiles C.
For all c ∈ P , define B(c) to be the maximal set of agents such that
(i) cB(c) ≤ 1, i.e., the sum of claims of agents in B(c) does not exceed the estate, and
(ii) for all i, l ∈ N , if ci ≥ cl and l ∈ B(c), then i ∈ B(c), i.e., if agent l is a member of B(c), then
all agents with claims larger than or equal to cl are also members of B(c).
Let D(c) = {i ∈ N \ B(c) : for all l ∈ N \ B(c), ci ≥ cl }, i.e., if B(c) 6= N , then D(c) 6= ∅ contains
the set of agents that have the highest claim among the agents in N \ B(c). Note that D(c) = ∅ if
and only if cN ≤ 1. Finally, let A(c) = B(c) ∪ D(c).
Roughly speaking, rule R works as follows. For claim profiles in ∪4j=0 C j , we specify awards to
agents according to their claims being larger than H or not. For claim profiles in P , rule R does
the following: it first ranks agents from highest claim to lowest claim. Then, to all agents in the
set B(c), R gives their full claim, and allocates the residual amount equally to agents in D(c). All
other agents receive 0.

1/4,
c ∈ C 4 ∪ C 0,




1/6,
ci ≤ H and c ∈ C 3 ,




1/2,
ci > H and c ∈ C 3 ,




1/3,
ci ≤ H and c ∈ C 2 ,



 1/6,
ci > H and c ∈ C 2 ,
Ri (c, 1) =
0,
ci ≤ H and c ∈ C 1 ,




1/3,
ci > H and c ∈ C 1 ,




ci ,
c∈P
and i ∈ B(c),



1−cB(c)

 |D(c)| , c ∈ P
and i ∈ D(c),



0,
c∈P
and i ∈
/ A(c).
We prove in Appendix A (Claim 1) that rule R satisfies efficiency, equal treatment of equals,
and claims monotonicity. Next, we show that the profile of claims c = (1, 1/3, 1/3, 1/3) is an
equilibrium for Γ(R) and since R(c) = (1/2, 1/6, 1/6, 1/6), we have a violation of equal division in
equilibrium.
Note that c ∈ C 3 . Then, a unilateral deviation by agent 1 can only result in a claim profile that
belongs to one of the sets C 3 , C 4 , or P , which induces the amounts (for agent 1) 1/2, 1/4, or 0,
respectively. Since at c agent 1 obtains 1/2, no unilateral deviation from c is beneficial for agent 1.
Next, a unilateral deviation by agent k ∈ {2, 3, 4} can only result in a claim profile that belongs to
one of the sets C 2 , C 3 , or P , which induces the amounts (for agent k) 1/6, 1/6, or 0, respectively.
Since at c agent k obtains 1/6, no unilateral deviation from c is beneficial for agent k.
Note that the rule described in Example 1 violates order preservation of awards (see Footnote 9)
and others oriented claims monotonicity (see Footnote 10). Furthermore, this rule is discontinuous
8
and it is an open problem if a continuous example can be constructed.
Finally, we show that equal division is restored in the equilibrium result of Theorem 2 for
more than three agents by adding a non-manipulation (or robustness) property: nonbossiness
(Satterthwaite and Sonnenschein, 1981) requires that no agent can change other agents’ awards by
changing his claim without changing his own award.
Nonbossiness: for all c ∈ C and all i ∈ N such that Ri (c) = Ri (c0i , c−i ), Rj (c) = Rj (c0i , c−i ) for
all j 6= i.
Theorem 3. Let rule R satisfy efficiency, equal treatment of equals, claims monotonicity, and
nonbossiness. Then,
(a) the maximal claims vector c̄ is a Nash equilibrium of the claim game Γ(R) and
(b) the outcome in all Nash equilibria of the claim game Γ(R) is min{k, En }1.
Proof.
(a) By claims monotonicity, for each agent i it is a weakly dominant strategy to claim c̄i . Hence,
c̄ is a Nash equilibrium of Γ(R).
(b) We first prove that equal treatment of equals, claims monotonicity, and nonbossiness imply
equal division for all Nash equilibria. Suppose that c is a Nash equilibrium of the claim game
Γ(R) and for some i, j ∈ N , (i) Ri (c) 6= Rj (c). Hence, by equal treatment of equals ci 6= cj .
Without loss of generality assume that ci < cj . Let c0i = cj and consider (c0i , c−i ). Since c is a
Nash equilibrium, Ri (c0i , c−i ) ≤ Ri (c). By claims monotonicity, Ri (c0i , c−i ) ≥ Ri (c). Hence, (ii)
Ri (c0i , c−i ) = Ri (c). Thus, by nonbossiness, (iii) Rj (c0i , c−i ) = Rj (c). Then, (i), (ii), and (iii) imply
Ri (c0i , c−i ) 6= Rj (c0i , c−i ); a contradiction to equal treatment of equals. Therefore, for all i, j ∈ N ,
Ri (c) = Rj (c), which proves an equal division vector is induced by all Nash equilibria. By efficiency,
this equal division vector equals min{k, En }1.
From the proof of Theorem 3 it becomes clear that even without efficiency Nash equilibria
outcomes respect equal division. However, without efficiency, some part of the estate might be
wasted.
Note that the rule described in Example 1 violates nonbossiness.
Remark 3. Independence of assumptions in Theorem 3
Note that claims monotonicity alone implies Theorem 2 (a). Hence, we show independence only
for Theorem 2 (b).
(i) The proof that it is essential that all agents have equal maximal claims is the same as in
Remark 1 (i).
(ii) As explained after the proof of Theorem 3, efficiency is only needed to obtain the efficient
equal division vector as equilibrium outcome. The following inefficient proportional rule satisfies
equal treatment of equals, claims monotonicity, and nonbossiness, but not efficiency: for all c ∈ C,
P 0 (c) = P (c; E/2), where P (c; E/2) denotes the outcome of the proportional rule where only half
the estate E2 is allocated.
(iii) To prove that equal treatment of equals is needed one can use the serial dictatorship rule as
described in Remark 1 (iii) – it satisfies efficiency, claims monotonicity, and nonbossiness but not
equal treatment of equals.
9
(iv) To prove that claims monotonicity is needed one can use rule R as described in Remark 1 (iv)
– it satisfies efficiency, equal treatment of equals, and nonbossiness, but not claims monotonicity.
(v) To prove that nonbossiness is needed one can use the same rule as in Example 1 – it satisfies
equal treatment of equals, and claims monotonicity, but not nonbossiness.
Remark 4. Divide-the-dollar games versus claim games
In the divide-the-dollar game (a simple version of Nash’s, 1953, demand game), two agents simultaneously submit their claims over a dollar. If the sum of claims does not exceed a dollar, each
agent receives his claim. Otherwise, both agents receive nothing. This simple version of the game
has infinitely many (pure strategy) Nash equilibrium outcomes, namely any division of the dollar.
Brams and Taylor (1994) and Anbarcı (2001) consider adaptations of the divide-the-dollar game
that exhibit equal division in Nash equilibrium.
Claim games can be considered as modified versions of the divide-the-dollar game (or Nash’s,
1953, demand game), the main difference being that agents are not punished as severely as they are
in divide-the-dollar games (i.e., receiving nothing) whenever the sum of claims exceeds the estate.
In a claim game, if the sum of claims does not exceed the value of the estate, all agents receive their
claims (as in a divide-the-dollar game). If, on the other hand, the sum of claims exceeds the value of
the estate, instead of not allocating the estate at all, a division rule with certain properties solves
the dispute over the estate and (in contrast to the divide-the-dollar game) an efficient outcome
is obtained. We consider entire classes of rules (determined by normative properties they share),
whereas most divide-the-dollar games use a fixed reference allocation (e.g., the zero share vector).
Similar to Brams and Taylor (1994) and Anbarcı (2001), our modifications induce equal division in
Nash equilibrium.
It is worthwhile mentioning that the rules we use for claim games satisfy most properties that
Brams and Taylor (1994) require for reasonable payoff schemes: (i) equal claims are treated equally,
(ii) no agent receives more than what he claimed, (iii) if the sum of claims does not exceed the
estate, then every agent receives his claim, (iv) if the sum of claims exceed the estate, nevertheless,
the whole estate is allocated, and (v) if all claims are higher than En , then the highest claimant
does no better than the lowest claimant (our rules only might fail to satisfy property (v)). Hence,
our results provide an alternative answer to the question “Can one alter the payoff structure of
the divide-the-dollar game in a reasonable way so that the egalitarian outcome is a noncooperative
solution of the corresponding game?” raised by Brams and Taylor (1994).
Remark 5. (Strong) Nash equilibria for Γ(P ), Γ(CEA) and Γ(CEL)
The proportional rule, the constrained equal awards rule, and the constrained equal losses rule
satisfy all properties introduced in this article. Hence, for these rules, claiming the largest possible
amount is always an equal-division Nash equilibrium. For the proportional rule and the constrained
equal losses rule, this is the unique Nash equilibrium of the associated claim game. However, if
agents are allowed to claim more than an equal share of the estate, the constrained equal awards
rule admits multiple (in fact infinitely many) equal-division Nash equilibria. This difference stems
from the fact that under the proportional rule and the constrained equal losses rule, claiming the
whole estate is a strictly dominant strategy for all agents whereas under the constrained equal
awards rule, it is a weakly dominant strategy.
Note that all Nash equilibria for the proportional rule, the constrained equal awards rule, and
the constrained equal losses rule are also strong (i.e., there exist no coalition that can make each of
its members strictly better off using a joint deviation). However, this is not a general result: there
10
exist rules satisfying equal treatment of equals, efficiency, order preservation of awards, and claims
monotonicity such that the corresponding claim game has Nash equilibria that are not strong (an
example is available from the authors upon request).
4
Concluding remarks
We analyze situations where an estate should be distributed among a set of agents, but claims to the
estate are impossible or difficult to verify. We model a simple and intuitive claim game where, given
the estate and a (estate division) rule satisfying some basic properties, agents simply announce their
claims. Our game can be thought of as a modified Nash demand game (or divide-the-dollar game)
where players are punished less severely than in the standard version of the game. Our results show
that first of all, claiming the largest possible amount is always a Nash equilibrium. Of course, this is
an intuitive and not very surprising result. However, in addition, we show that even though we do
not focus on any specific outcome function to be used in our claim game, equal division is the unique
Nash equilibrium outcome. Since most well-known rules satisfy all the properties we require (e.g.,
the proportional rule, the constrained equal awards rule, and the constrained equal losses rule), our
results can be interpreted as a non-cooperative support for equal division in estate division conflicts.
The advantage of this implementation result is that equal division is the outcome that is based on
the (strategic) choices of the agents rather than a fixed outcome that is externally imposed; given
the desirable properties of the underlying rule this process might therefore be considered fair and its
outcome thus acceptable. Finally, future research on this topic might analyze situations in which
partial verification is possible and agents spend resources to support their claims (e.g., hiring a
lawyer in a court case).
Appendix A
Claim 1. Rule R as defined in Example 1 satisfies equal treatment of equals, efficiency, and claims
monotonicity.
Proof. Equal treatment of equals follows immediately from the definition of rule R.
P
j , c ≥ 1 and
Efficiency: Note that for all c ∈ ∪4j=0 CP
Rl (c) = P
1. Assume thatP
c ∈ P . If cN ≤ 1,
N
P
then R(c) = c. Finally, if cN > 1, then Rl (c) = i∈B(c) Ri (c)+ i∈D(c) Ri (c)+ i∈N \A(c) Ri (c) =
P
1−cB(c)
cB(c) + i∈D(c) |D(c)|
+ 0 = 1.
Claims Monotonicity: Let i ∈ N , c = (ci , c−i ), and c0 = (c0i , c−i ) such that ci < c0i . We show that
Ri (c) ≤ Ri (c0 ) for the following (exhaustive) cases.
Case 1 : c, c0 ∈ P .
If i ∈
/ A(c), then Ri (c) = 0 ≤ Ri (c0 ). If i ∈ A(c), then i ∈ A(c0 ), i.e., i ∈ B(c0 ) or i ∈ D(c0 ). If
i ∈ B(c0 ), then Ri (c) ≤ ci ≤ c0i = Ri (c0 ) and we are done. Assume that i ∈ D(c0 ). Since ci < c0i , for
all j ∈ N \B(c), cj < c0i . Hence, A(c0 )\{i} ⊆ B(c). Therefore, for all j ∈ B(c0 ), Rj (c0 ) = Rj (c), and
for all j ∈ D(c0 ) \ {i}, Rj (c0 ) ≤ Rj (c). Thus, we showed that for all j ∈ A(c0 ) \ {i}, Rj (c0 ) ≤ Rj (c).
Since for all j ∈ N \ A(c0 ), Rj (c0 ) = 0 and R is efficient it follows that Ri (c) ≤ Ri (c0 ).
Case 2 : c ∈ P and for some j ∈ {0, 1, 2, 3, 4}, c0 ∈ C j .
P
Note that L(c) = {i}
N \ {i}. Since L ≥ 1/3, cN \{i} ≥ 1 and l6=i Rl (c) = 1.
P and LH(c) ∪ H(c) =
Thus, Ri (c) = 1 − l6=i Rl (c) = 0 ≤ Ri (c0 ).
11
Case 3 : for some j ∈ {0, 1, 2, 3, 4}, c ∈ C j and c0 ∈ C j .
Note that either [i ∈ LH(c) and i ∈ LH(c0 )] or [i ∈ H(c) and i ∈ H(c0 )]. Thus, Ri (c) = Ri (c0 ).
Case 4 : for some j ∈ {1, 2, 3, 4}, c ∈ C j and c0 ∈ C j−1 .
Note that i ∈ LH(c) and i ∈ H(c0 ). Then, by the definition of R, for j = 1, 4, Ri (c) < Ri (c0 ) and
for j = 2, 3, Ri (c) = Ri (c0 ).
Appendix B
In this appendix we describe what happens if we require that the estate E is always completely
allocated among the agents. Formally, a (full division) rule is a function P
R : C → RN
+ that
N
associates with each claims vector c ∈ C an awards vector x ∈ R+ such that
xi = E.
Note that Lemma 1 does not hold anymore, i.e., it is not always the case that in every Nash
equilibrium c of the claim game Γ(R), cN ≥ E; e.g., for the constant rule that always assigns E/n
to each agent, every claims vector is a Nash equilibrium. Although this lemma is used in some of
our proofs, it is used only in order to show that in equilibrium the entire estate is allocated. Hence,
the fact that the whole estate is always allocated can be used instead of Lemma 1. Furthermore, all
results that state that the division vector in a Nash equilibrium is min{k, En }1 are changed to have
the En 1 division vector. To summarize, Theorems 1, 2, and 3 hold with minimal changes in the
statements and proofs. Finally, the only adjustment of Example 1 needed to fit the model described
here is to change rule R in Example 1 to rule R̃ as follows: for every c ∈
/ P let R̃(c) = R(c) and for
E
every c ∈ P let R̃(c) = 4 .
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